Maclaurin series of eˣ, sin x, cos x, ln(1+x), (1+x)ⁿ — derive all: Visual walkthrough

4.3.17 · D2 · Maths › Calculus III — Sequences & Series › Maclaurin series of eˣ, sin x, cos x, ln(1+x), (1+x)ⁿ — deri

Step 1 — Curve kya hoti hai, aur polynomial kya hoti hai?

KYA. Ek function $f$ ek rule hai jo ek number $x$ leta hai aur height $f (x)$ wapas deta hai. Un saari heights ko draw karo toh ek curve milti hai. Ek polynomial ek special, seedha-saadha curve hai jo sirf $x$ ki powers jodhke banti hai: $P (x) = constant a_{0} + ek line a_{1} x + ek parabola a_{2} x^{2} + ek S-bend a_{3} x^{3} + \dots$

$a_{0}$ ek plain number hai — yeh poori curve ko bas upar ya neeche uthata hai.
$a_{1} x$ use tilt karta hai (ek seedha ramp).
$a_{2} x^{2}$ use ek bowl mein bend karta hai.
$a_{3} x^{3}$ ek S-shape add karta hai, aur isi tarah aage bhi.

Har $a_{k}$ ek dial hai. Dial ghumao toh shape ki ek feature badal jaati hai. Humara poora kaam yeh hai ki woh dial settings dhundho jo tame polynomial ko ek wild function ki nakal karayen.

YEH YAHAN SE KYUN. Polynomials hi ek maatra curves hain jinhe hum thodi si numbers se poori tarah control kar sakte hain. Sine, cosine, logs — yeh sab mushkil hain. Toh hum poochhte hain: kya hum ek mushkil curve ko un dials se fake kar sakte hain jo hum samajhte hain?

PICTURE. Dekho ki har power apni alag personality shape mein kaise add karti hai.

Figure — Maclaurin series of eˣ, sin x, cos x, ln(1+x), (1+x)ⁿ — derive all

Step 2 — Meeting point chunna: $x = 0$

KYA. Hum ek khaas input, $x = 0$ , choose karte hain, aur demand karte hain ki hamari polynomial real function se wahan agree kare — sirf height mein nahi, balki behaviour ki har layer mein.

$x = 0$ HI KYUN? Kyunki $x = 0$ $x$ ki powers ko ek bahut clean tarike se khatam kar deta hai: $0^{1} = 0, 0^{2} = 0, 0^{3} = 0, \dots$ Jab hum $x = 0$ plug karte hain, toh har term jisme $x$ hai woh gaayab ho jaati hai. Yahi collapse poora trick hai — yeh humein ek baar mein ek dial peel off karne deta hai. (Kisi doosre point $a \neq = 0$ pe milna bhi allowed hai — woh Taylor series (general centre a) hai — lekin $0$ sabse friendly hai.)

PICTURE. Real curve $f$ aur hamari polynomial $f$ ko ek akele point $(0, f (0))$ pe touch karne pe majboor kiya ja raha hai.

Step 3 — Height match karo: $a_{0}$ dhundho

KYA. Polynomial mein $x = 0$ daalo: $P (0) = a_{0} + a_{1} \cdot 0 + a_{2} \cdot 0^{2} + \dots = a_{0} .$

$a_{0}$ ke alawa har dial $0$ ki kisi power se multiply ho raha hai, isliye woh nikal jaata hai. $P (0) = f (0)$ set karne se milta hai: $a_{0} = f (0)$

$a_{0}$ — height dial, ab meeting point pe $f$ ki sacchi height se lock ho gaya.
$f (0)$ — curve ki $x = 0$ pe actual value.

KYUN. Yeh sabse crude possible fake hai: bilkul sahi height pe ek flat horizontal line. Yeh $f$ ko ek point pe touch karti hai lekin yeh nahi dekhti ki $f$ chadh rahi hai ya utar rahi hai.

PICTURE. Hari flat line sirf height match karti hai — yeh turant dur ho jaayegi.

Step 4 — Slope match karo: derivative kya hoti hai, aur $a_{1}$ dhundho

KYA hai derivative? Ek point pe curve ki slope yeh hai ki woh kitni steep hai — ek kadam sideways lene pe tumhe kitni height milti hai. Yeh measure karne ka tool derivative hai, jise $f^{'} (x)$ likha jaata hai. Socho curve pe zoom in karo jab tak woh seedhi na lage; woh choti seedhi tilt hi $f^{'} (x)$ hai.

YEH TOOL HI KYUN, KOI AUR KYUN NAHI? Sirf height (Step 3) ek flat line deti hai. Yeh bhi capture karne ke liye ki curve kis taraf ja rahi hai, humein woh ek tool chahiye jo steepness report kare — woh exactly derivative hai. Koi doosri operation yeh nahi batati ki "yahan kitna teda hai?"

KAISE. Ek power ko differentiate karne se uska exponent ek se kam ho jaata hai aur purana exponent se multiply ho jaata hai: $\frac{d}{d x} x^{k} = k x^{k - 1} .$ Isse poori polynomial pe apply karo: $P^{'} (x) = a_{1} + 2 a_{2} x + 3 a_{3} x^{2} + \dots$ Ab $a_{0}$ term gaayab ho gayi (ek constant ki derivative $0$ hoti hai — ek flat line mein koi tilt nahi hoti). $x = 0$ daalo: bacha hua har $x$ mar jaata hai, sirf yeh bachta hai: $a_{1} = f^{'} (0)$

$a_{1}$ — tilt dial.
$f^{'} (0)$ — $x = 0$ pe curve ki sacchi slope.

Ab hamara fake ek tangent line hai: sahi height aur sahi tilt.

PICTURE. Line ab $0$ pe curve ke saath exactly jhukti hai.

Step 5 — Bend match karo: $n!$ yahan se paida hota hai, $a_{2}$ dhundho

KYA. Tangent line seedhi hai; real curve usse dur bend ho jaati hai. Bending ko derivative of the derivative se measure kiya jaata hai, jise $f^{''} (x)$ likha jaata hai — "slope khud kitni tezi se badal rahi hai".

KYUN. Ek line height aur tilt toh match kar sakti hai lekin bend kabhi nahi. Ek curved shape ko hug karne ke liye humein curvature match karni hogi, aur second derivative exactly curvature-reporter hai.

KAISE — aur dekho factorial kaise appear hota hai. $P^{'} (x)$ ko phir se differentiate karo: $P^{''} (x) = 2 a_{2} + (3 \cdot 2) a_{3} x + (4 \cdot 3) a_{4} x^{2} + \dots$ Dhyaan do ki $x^{2}$ term ko do baar hit kiya gaya: pehle $x^{2} \to 2 x$ , phir $2 x \to 2$ . Do derivatives ne numbers $2$ aur $1$ ko saath multiply kiya — woh $2! = 2$ hai. $x = 0$ daalo: $P^{''} (0) = 2 a_{2} = f^{''} (0) ⟹ a_{2} = \frac{f ^{''} ( 0 )}{2 !}$

$2!$ decoration nahi hai — yeh $x^{2}$ ko do baar differentiate karne se bachne wala number hai.
Generally, $x^{n}$ ko poore $n$ baar differentiate karne se $n \cdot (n - 1) \dots 2 \cdot 1 = n!$ milta hai, aur baaki har term ya toh gaayab ho jaati hai (chhoti powers) ya abhi bhi $x$ carry kar rahi hoti hai (badi powers, jo $0$ pe mar jaati hain). Isliye: $a_{n} = \frac{f ^{(n)} ( 0 )}{n !}$

Yeh master coefficient rule hai — poora engine, ab dekha gaya, sirf bataya nahi.

PICTURE. Dekho $x^{n}$ ko $n$ baar differentiate hote aur numbers $n, n - 1, \dots, 1$ ko $n!$ mein pile up hote.

Step 6 — $e^{x}$ ko engine mein daalo

KYA. Ab sabse friendly possible function lo: $f (x) = e^{x}$ , woh curve jiski slope har jagah apni height ke barabar hoti hai, isliye $\frac{d}{d x} e^{x} = e^{x}$ .

$e^{x}$ PEHLE KYUN. Kyunki $e^{x}$ ki har derivative phir se $e^{x}$ hai, aur $e^{0} = 1$ . Isliye har numerator $f^{(n)} (0)$ wahi number hai, $1$ . Engine ko zyada kaam hi nahi karna padta: $a_{n} = \frac{f ^{(n)} ( 0 )}{n !} = \frac{1}{n !} .$

KAISE. Pieces stack karo: $e^{x} = 1 + x + \frac{x ^{2}}{2 !} + \frac{x ^{3}}{3 !} + \dots = n = 0 \sum \infty \frac{x ^{n}}{n !}$

$1$ — height match karta hai ( $e^{0} = 1$ ).
$x$ — slope match karta hai ( $1$ ).
$\frac{x ^{2}}{2 !}$ — bend match karta hai, factorial se divide kiya taaki woh haawi na ho jaaye.
Har extra term chhota hota hai kyunki $n!$ explosively badhta hai, isliye kuch terms hi $0$ ke paas curve ko hug karne ke liye kaafi hain. (Yeh actually sare $x$ ke liye hug karta hai — dekho Radius & interval of convergence.)

PICTURE. Ek ek karke terms add karo aur dekho polynomial $e^{x}$ ke around wrapte hue baahir creep karti hai.

Step 7 — Wahi engine, char aur functions

KYA. Kuch bhi naya invent nahi karna — bas wahi crank alag derivative patterns ke saath ghumaao $0$ pe.

$sin x$ : derivatives cycle karte hain $sin \to cos \to - sin \to - cos$ , $0$ pe values dete hain $0, 1, 0, - 1, \dots$ . Sirf odd powers bachti hain, signs alternate karti hain: $sin x = x - \frac{x ^{3}}{3 !} + \frac{x ^{5}}{5 !} - \dots$
$cos x$ : wahi cycle $cos 0 = 1$ se shuru hoti hai: values $1, 0, - 1, 0, \dots$ . Sirf even powers bachti hain: $cos x = 1 - \frac{x ^{2}}{2 !} + \frac{x ^{4}}{4 !} - \dots$
$ln (1 + x)$ : ek Geometric series integrate karna zyada easy hai: $\frac{1}{1 + x} = 1 - x + x^{2} - \dots$ term-by-term integrate hoke deta hai: $ln (1 + x) = x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \dots (- 1 < x \leq 1)$
$(1 + x)^{n}$ : derivatives $n, n (n - 1), \dots$ factors peel off karti hain, binomial series deti hain: $(1 + x)^{n} = 1 + n x + \frac{n ( n - 1 )}{2 !} x^{2} + \dots$

YEH SAATH KYUN. Yeh sirf numerators $f^{(n)} (0)$ mein alag hain. Neechey $n!$ , $x^{n}$ , " $0$ pe match" ka idea — sab identical.

PICTURE. Ek machine, char output belts.

Ek-picture summary

Upar wali akeli figure sab kuch compress karti hai: point $x = 0$ chuno → height, phir slope, phir bend, phir bend-of-bend match karo → har match ek dial force karta hai $a_{n} = f^{(n)} (0) / n!$ → growing polynomial har term ke saath sacchi curve ko tightly hug karti hai.

Recall Feynman retelling — poora walkthrough seedhe shabdon mein

Socho tum ek pahadi raaste pe ek jagah khade ho ( $x = 0$ ). Tum poora raasta nahi dekh sakte, lekin tum feel kar sakte ho char cheezein: tum kitne upar ho, tum kitni steep taraf jhuke ho, raasta kitna curve kar raha hai, aur woh curve khud kitni tezi se badal rahi hai. Turns out yeh feelings tumhare paas ki road redraw karne ke liye kaafi hain. Toh hum simple pieces se ek fake road banate hain: ek flat block (height), ek ramp (slope), ek gentle bowl (bend), ek S-wiggle (bend-of-bend), aur isi tarah aage bhi. Hum har piece ko tune karte hain taaki woh exactly ek feeling match kare — pehle height, phir tilt, phir curve. Har naya piece jo hum add karte hain use deliberately ek factorial se divide karke chhhota kiya jaata hai, taaki badi shapes chhoti corrections pe stampede na karein. Jab function $e^{x}$ hoti hai, toh har feeling wahi number hai ( $1$ ), isliye pieces sirf $1, x, x^{2} /2, x^{3} /6, \dots$ hain. Sine aur cosine $0, 1, 0, - 1$ ka repeating pattern feel karte hain, isliye unke pieces powers skip karte hain aur sign flip karte hain. Bas itna hi hai — ek Maclaurin series bas tuned shapes ka ek stack hai jo copy karti hai ki curve ek point pe kaise feel hoti hai.

Recall

$x = 0$ daalne se ek waqt mein ek coefficient isolate kyun hota hai? ::: Kyunki abhi bhi $x$ carry kar raha har term $0$ ban jaata hai, isliye sirf us derivative ki constant term bachti hai. $a_{n}$ mein $n!$ kahaan se aata hai? ::: $x^{n}$ ko exactly $n$ baar differentiate karne se: $\frac{d ^{n}}{d x ^{n}} x^{n} = n!$ . $e^{x}$ sabse easy series kyun hai banana? ::: $e^{x}$ ki har derivative $e^{x}$ hai aur $e^{0} = 1$ , isliye har numerator $f^{(n)} (0) = 1$ hai.

Connections

Parent topic — taiyaar formulas
Taylor series (general centre a) — wahi machine, meeting point $a \neq = 0$
Geometric series — $ln (1 + x)$ ke peeche shortcut
Radius & interval of convergence — har series kitni door tak faithful rehti hai
Euler's formula — $e^{x}$ , $sin$ , $cos$ series combine karo
L'Hôpital & limits via series — series tricky $0/0$ limits solve karte hain
Binomial theorem (integer n) — $(1 + x)^{n}$ ka terminating case